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Sensitivity Analysis in Atmospheric Data Assimilation

Ron Errico, Ricardo Todling, Nate Winslow, Yanqiu Zhu

Ron GelaroNASA Global Modeling and Assimilation Office (GMAO)

Goddard Space Flight Center Greenbelt, MD USA

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ScatterometerSSM/I Ozone

ATOVS Satwinds Geostationary Radiances

Radiosondes Pilots and Profilers Aircraft

Synops and Ships Buoys

09–15 UTC5 September 2003

The Observing System

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The observing system

FVGSI 16-Jan-2003 00UTC All data: 1,178,200 observations

(x 100,000)8.44.2

Upper-air virtual temperature

Data Types

Brightness temperatureSurface (2m) pressure

Surface (10m) wind speedUpper-air specific humidity

Upper-air meridional windUpper-air zonal wind

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Status of satellite data usage in operational NWP

• Satellite observations account for ~90% of all data assimilated in operational data assimilation/forecast systems

• These data account for << 1% of the available observations from lower Earth-orbiting systems

• The gap between the number of available and assimilated satellite observations is likely to widen significantly over the next decade

⇒ Intelligent strategies for data selection and usage must be developed

⇒ Flexible and efficient tools for ascertaining the “importance” of observations are required...

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Adjoint sensitivity analysis

)( af xx M=1. Consider a model:and a (differentiable) scalar forecast measure: )( fxJJ =

3. The solution is:

jj

i

j

i xJ

x

x

xJ

fa

f

a ∂∂

∂

∂=

∂∂

∑

ixJ a/∂∂ such that:

iii

xxJJ aa

′∂∂

= ∑δ )()( xxx JJJ −′+=Δapproximates

2. Determine

…Estimate the response of model output to possible perturbations of model input

orf

T

a xM

x ∂∂

=∂∂ JJ

Sensitivity of J w.r.t the analysis Adjoint model (transpose of M)

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Example adjoint sensitivity calculation

FVGCM 24hr Forecast Error Sensitivity 00Z 15 Jan 2004

aθ∂∂J (shaded) aθ basic state (contour)

FVGCM adjoint model with simple physics

24T242

1 eEe=J

Applications: predictability, system monitoring, parameter estimation, adaptive observing…

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1-Tb

Tb )( RHHPHPK +=

bbba )()( xKHIKyHxyKxx −+=−+=

( ))( bba xyKxx H−+=Analysis problem:

where

Linear analysis problem:

Ta Kyx

=∂∂

b1-T

bT )( HPRHHPK +=

Sensitivity of the analysis to the observations:

where

Data assimilation adjoint theory (Baker and Daley, 2000)

See also Le Dimet et al. 1995, Fourrie et al. 2002, Cardinali 2004(?)

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We have the sensitivity of the forecast measure to the analysis:

Now, relate the sensitivity of the forecast measure to the observations:

f

T

a xM

x ∂∂

=∂∂ JJ

…and the sensitivity of the analysis to the observations:

Ta Kyx

=∂∂

Combine (1) and (2) to get the sensitivity of the forecast measure to the observations:

a

axy

xy ∂

∂∂∂

=∂∂ JJ

f

TTx

MKy ∂

∂=

∂∂ JJ

(observation space) (model space)

or

(2)

(1)

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Adjoint of Forecast and Assimilation ProcedureAdjoint of Forecast and Assimilation Procedure

Forecast Model

Forecast(xf)

Analysis(xa)

Adjoint Data Assimilation

System

Observation Impact What is the impact of the observations on reducing forecast error?

AnalyisSensitivity

(∂J/∂xa)

AdjointForecast

ModelObservationSensitivity

(∂J/∂y)

BackgroundSensitivity

(∂J/∂xb)

Observations(y)

Data Assimilation

System

Background(xb)

Assimilation and Forecast ProcedureAssimilation and Forecast Procedure

“End-to-End” Sensitivity in a Data Assimilation-Forecast System

Grad FcstMeasure(∂J/∂xf)

Define FcstMeasure J(xf)

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Impact of observations on forecast error

The forecast error difference, , is due entirely to the assimilation of observations at 00UTC

30243024 eee Δ=−

We seek an estimate of in terms of sensitivity gradients in observation space…

3024eΔ⇒

Langland and Baker (2004)

OBSERVATIONS ASSIMILATED AT

00UTC

+24h

30e24e

axbx

vx

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Observation impact equation (Langland and Baker, 2004)

1. Define the energy-weighted scalar error, e.g.:

)()( v24T

v2424 xxExx −−=e

2421

24 eJ =and forecast measure:

(observation space)

30

30

24

243024

3024 ),(

xxxx

∂∂

+∂∂

−=ΔJJe

2. Express in terms of gradients at verification time (exact!):3024eΔ

3. Apply then to obtain the approximate expression at initial time:

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

−=≈Δb

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a

24Tb

3024

3024 ),(

xxKHxy JJee δ

TM TK

(model space)

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NAVDAS: NRL Atmospheric Variational Data Assimilation System (3D, 1o lat-lon, 60 levels)

NOGAPS: Navy Operational Global Atmospheric Prediction System (T79L30)

The analysis procedure:

The forecast model:

⇒ Adjoint excludes moist physics

⇒ Adjoint excludes moist observations, ozone

Results from Langland and Baker, 2004

Diagnosis of observation impact on short-range forecast errors in the Navy’s operational DAS/forecast system

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Rawinsonde Profiles

Decrease Increase

00UTC 10 Dec 2002

Total Impact = -2.05 J/kg

Langland and Baker, 2004

Observation impact on 3024 ee −

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ATOVS Temperature Retrievals

Total Impact = -1.06 J/kg

Langland and Baker, 2004

Decrease Increase

00UTC 10 Dec 2002

Observation impact on 3024 ee −

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Commercial Aircraft Observations

Total Impact = -0.36 J/kg

Langland and Baker, 2004

Decrease Increase

00UTC 10 Dec 2002

Observation impact on 3024 ee −

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-100

-80

-60

-40

-20

0

20

40 SHEMNHEM

RAOB

ATOVS

SATW

LAND SHIPAIRW

AUSN

3,000,0002,000,0001,000,000

Ob count

J kg-1

June and December 2002

Number of data assimilated at 00UTC

Impact of data on

Observation impact – by hemisphere

Langland and Baker, 2004

3024 ee −

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Large Impact of Observations in Cloudy Regions

0

0.001

0.002

0.003

0.004

0.005

0 10 20 30 40 50 60 70 80 90

Cloud Cover (% )

Ob

Impa

ct

Fig. 4: Observation impact (average magnitude perobservation, in J kg-1) as a function of model-diagnosed cloud-cover. The “impact” in this figure includes both improvements and degradations of 72h global forecast error. Based on results from 29 June – 28 July 2002.

SATWINDRAOBATOVS

Observation impact – cloud cover correlation

…large impact of observations in cloudy regions….both positive and negative...

Langland and Baker, 2004

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Observation impact – assimilation system tuning

N. Baker, NRL

Error Reduction (J kg-1)

Global Domain

4842eδ

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December 2002

J kg -1

nonlinear model forecast differences

estimate using dry adjoint procedure

explains 75% of

• moist observations?

• adjoint model physics?

• nonlinearity?

Remaining 25 %:

Accuracy of observation impact estimate

3024eδ 30

24eΔ3024eδ

3024eΔ

Langland and Baker, 2004

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Nonlinear analysis problems: GSI

))(())(()( 1T1T dxRdxxBxx −−+= −− δδδδδ HHL

21)()( qq LLL ++= xx δδLMinimize:

⎩⎨⎧

<

≥=

0

002T

11 qq

qLq ρ

…penalty for negative humidity

⎪⎩

⎪⎨⎧

>−

≤=

satsat

satq qqqq

qqL 2T

22 )(

0

ρ…penalty for super saturation

…Consider the incremental variational problem solved in the NCEP grid-point statistical interpolation (GSI) scheme

Development of adjoint GSI requires development of tangent linear algorithm as first step…

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GSI TLM

Δ GSI

Perturb: all u,v (O-F x 0.1)Response: Tv (level 10)

Tangent linear behavior of the GSI algorithm

100 iterations

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Δ GSI

Perturb: all Rad (O-F x 0.1)Response: v (level 20)

100 iterations

Tangent linear behavior of the GSI algorithm

GSI TLM

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Tangent linear behavior of the GSI algorithm

Ratio = rms (TLM)rms (Δ GSI)

q-penalty OFF

Pert = all u,v (O-F x 0.01)

q-penalty ONlevel=15, outerloop=1

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Δ GSI

Perturb: all u,v (O-F x 0.01)Response: u (level 15)

Tangent linear behavior of the GSI algorithm

80 iterations

q-penalty ON

GSI TLM

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Δ GSI

Perturb: all u,v (O-F x 0.01)Response: u (level 15)

Tangent linear behavior of the GSI algorithm

80 iterations

q-penalty OFF

GSI TLM

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Effect of q-penalty on GSI increments

q-inc q-penalty ON Δq-inc (q-penalty ON – q-penalty OFF)

Specific humidity, 00Z 19Jun2004, level=10, outerloop=1, iteration=100

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Effect of q-penalty on GSI increments

Virtual Temperature, 00Z 19Jun2004, level=10, outerloop=1, iteration=100

TV-inc q-penalty ON ΔTV-inc (q-penalty ON – q-penalty OFF)

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Status of GSI adjoint development

• Tangent linear version of GSI (autumn release) has been developed and tested extensively for a range of perturbation types...major step toward generation of adjoint.

• Adjoint testing planned for early summer

• Tangent linear system is well behaved (representative of GSI) overall:- good to excellent results for perturbed observations of u, v, T, q, radiance- results improve, especially for q, when penalties for negative moisture/super saturation are removed or reduced- response to ozone perturbations poor (highly nonlinear), but significance is unclear

• Update to March release of GSI underway:- variational QC- begin consolidation of nonlinear and tangent codes

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General conclusions…so far…

• Adjoint forms of a model and assimilation system allow efficient estimation of analysis and/or observation sensitivity (impact)

- determined with respect to observational data, background fieldsor assimilation parameters, all computed simultaneously

• Sensitivity information should be useful for designing intelligent data selection strategies, and possibly guide future observing system design

- permits arbitrary aggregation of sensitivities, e.g., by data type, channel, location, etc.- information obtained should compensate pain of initial development

• Provide unique insight into model/assimilation system behavior and design

• Complement, but not replace, traditional techniques for estimating observation impact (OSEs, OSSEs)

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1st order cience questions…

• Trade-offs between assimilating fewer observations with greater individual sensitivity, or more observations with less individual sensitivity

• How best to use observations in cloudy regions

• Whether/how sensitivity information can be utilized effectively for system tuning, e.g., observation error, bias estimation parameters,…

• Under what circumstances the required assumptions and simplifications render results unusable (simplified physics, linearity)

• Whether extremely nonlinear “switches” in forward systems are either necessary or appropriate